Non-linear LinesHow to Avoid Lying, without Confusing.

[Bacle]

"Non-linear Lines". How to Avoid Lying, without Confusing.

Hi, everyone:

I will be teaching an intro course in Linear Algebra this Spring.

Problem I am having is that the definition of linear does not

apply to lines that do not go through the origin:

Let L:x-->ax+b

Then L(x+y)=ax+ay+b =/ L(x)+L(y)

similarly: L(cx)=acx+b =/ c(L(x))=cax+cb

Which is true only for c=0 . So lines are affine objects, carelessly described as linear,
as in 'linear equations'

So, how does one reasonably avoid bringing up the issue of affine vs. linear
and still not refer to a collection of equations

ax_i +b=0

as linear equations?

 

[Fredrik]

Is this really a problem? How about telling them something like this: Mathematicians have found it useful to have a term for maps that satisfy T(ax+by)=aTx+bTy. You (the students) will understand why before this course is over. The term they chose for such maps is "linear". It would make just as much sense to use that term for maps that satisfy T(ax+by)=aTx+bTy+z for some z, because all such maps take straight lines to straight lines. The problem is, if we call the members of this larger set of maps "linear", then what do we call "linear maps with z=0"? "Linear and origin-preserving"? This gets annoying pretty quickly, so we just call them "linear".


Student: OK, but then what do we call the maps with arbitrary z?

Teacher: **** you!

 

[Studiot]

I'm glad someone is giving this topic consideration.

Let us generalise (as mathematicians are won't to do).

Thre are many examples where less than perfect terminology has arisen, become entrenched and become the source of confusion.
All too often this is hidden by promoting a simplified definition, hammered into earlier years students, who then find it difficult to 'unlearn' the half truth in order to embrace the wider picture.

Why not simply be up front about this and explain that there is some unfortunate terminology so careful attention needs to be placed upon the boundaries within which your definitions apply.

In all conscience students who go on to greater things will meet plenty of such examples in their future studies. The rest need not worry, but simply stay within the guidelines.

 

[atyy]

Just say there are (at least) two different definitions of "linear".

Even the OED gives multiple definitions of words.

And that in this course, we use this particular definition throughout.

 

[blue_raver22]

I would respond by explaining the difference between linear and affine objects. Linear objects, such as lines that go through the origin, follow the properties of linearity and can be described by linear equations. However, affine objects, such as lines that do not go through the origin, do not follow these properties and cannot be described by linear equations.

To avoid confusion, it would be important to clearly define and distinguish between linear and affine objects in the course. This can be done by introducing the concept of an affine space, which is a generalization of a vector space that includes affine objects. This will help students understand that while lines may be described as linear in everyday language, they are actually affine objects in mathematics.

Furthermore, to avoid lying to students, it is important to use accurate terminology and not refer to affine objects as linear. This can be achieved by using terms such as "affine lines" or "affine equations" when discussing non-linear lines. It may also be helpful to provide examples and visual aids to help students understand the concept of affine objects.

Overall, it is important to be clear and precise in teaching about non-linear lines and to avoid confusing students by using inaccurate terminology. By properly defining and distinguishing between linear and affine objects, students can better understand the properties and characteristics of non-linear lines.